Problem: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}x-2y+3z&=6 \\-4x+5y-6z&=8 \\7x-8y+9z&=1\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Answer: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}x-2y+3z&=6 \\-4x+5y-6z&=8 \\7x-8y+9z&=1\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{1}x+({-2})y+{3}z&=6 \\{-4}x+{5}y+({-6})z&=8 \\{7}x+({-8})y+{9}z&=1\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {1} & {-2} & {3} \\ {-4} & {5} & {-6} \\ {7} & {-8} & {9} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {1} & {-2} & {3} \\ {-4} & {5} & {-6} \\ {7} & {-8} & {9} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 6 \\ 8 \\ 1 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 1 & -2 & 3 \\ -4 & 5 & -6 \\ 7 & -8 & 9 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 6 \\ 8 \\ 1 \end{array} \right]$